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\hypersetup{
  pdftitle={寿险精算现值总结},
  pdfauthor={滕帆},
  hidelinks,
  pdfcreator={LaTeX via pandoc}}

\title{寿险精算现值总结}
\subtitle{保险精算学}
\author{滕帆}
\date{2023-09}
\institute{浙大宁波理工学院金融学系}

\begin{document}
\frame{\titlepage}

\begin{frame}[allowframebreaks]
  \tableofcontents[hideallsubsections]
\end{frame}
\hypertarget{ux9669ux79cdux4ebaux5bffux4fddux9669ux548cux5e74ux91d1}{%
\section{险种：人寿保险和年金}\label{ux9669ux79cdux4ebaux5bffux4fddux9669ux548cux5e74ux91d1}}

\begin{frame}{险种：人寿保险和年金}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
人寿保险

\begin{itemize}
\tightlist
\item
  定期寿险
\item
  终身寿险
\item
  两全保险
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
年金

\begin{itemize}
\tightlist
\item
  定期年金
\item
  终身年金
\item
  延迟年金
\end{itemize}
\end{column}
\end{columns}
\end{frame}

\hypertarget{ux5bffux9669ux7cbeux7b97ux73b0ux503c}{%
\section{寿险精算现值}\label{ux5bffux9669ux7cbeux7b97ux73b0ux503c}}

\begin{frame}{精算符号}
\protect\hypertarget{ux7cbeux7b97ux7b26ux53f7}{}
\[
\begin{aligned}
\Ax{\termxn}\\
\Ax*{\termxn}\\
\Ax{x}\\
\Ax{\endowxn}\\
\end{aligned}
\]
\end{frame}

\begin{frame}{身故年末给付的定期寿险}
\protect\hypertarget{ux8eabux6545ux5e74ux672bux7ed9ux4ed8ux7684ux5b9aux671fux5bffux9669}{}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{itemize}
\tightlist
\item
  保险责任：身故
\item
  保险期限：n年
\item
  投保年龄：x岁
\item
  给付方式：身故年末给付
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
\[
\Ax{\termxn}
\]
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{itemize}
\tightlist
\item
  保险责任：身故
\item
  保险期限：n年
\item
  投保年龄：x岁
\item
  给付方式：身故年末给付
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
\[
\begin{aligned}
\Ax{\termxn} =& \qx{x} v+\qx[1|]{x} v^2 \\
&+ \ldots + \qx[n-1|]{x} v^n
\end{aligned}
\]
\end{column}
\end{columns}
\end{frame}

\begin{frame}{即死即付的定期寿险}
\protect\hypertarget{ux5373ux6b7bux5373ux4ed8ux7684ux5b9aux671fux5bffux9669}{}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{itemize}
\tightlist
\item
  保险责任：身故
\item
  保险期限：n年
\item
  投保年龄：x岁
\item
  给付方式：身故即付
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
\[
\Ax*{\termxn}
\]
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{itemize}
\tightlist
\item
  保险责任：身故
\item
  保险期限：n年
\item
  投保年龄：x岁
\item
  给付方式：身故年末给付
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
\[
\begin{aligned}
&\Ax*{\termxn}\\
&=\int_0^n{v^t \px[t]{x}\mu_{x+t}dt}
\end{aligned}
\]
\end{column}
\end{columns}
\end{frame}

\begin{frame}
不加证明地给出身故年末给付定期寿险和即死即付定期寿险的精算现值之间关系:

在身故人数均匀分布条件下

\[
\Ax*{\termxn}=\frac{i}{\delta}\Ax{\termxn}
\]
\end{frame}

\hypertarget{ux5e74ux91d1ux7cbeux7b97ux73b0ux503c}{%
\section{年金精算现值}\label{ux5e74ux91d1ux7cbeux7b97ux73b0ux503c}}

\begin{frame}{精算符号}
\protect\hypertarget{ux7cbeux7b97ux7b26ux53f7-1}{}
\[
\begin{aligned}
\ax**{\endowxn}\\
\ax**{x}
\end{aligned}
\]
\end{frame}

\begin{frame}{年初给付的生命年金}
\protect\hypertarget{ux5e74ux521dux7ed9ux4ed8ux7684ux751fux547dux5e74ux91d1}{}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{itemize}
\tightlist
\item
  保险责任：生存
\item
  保险期限：n年
\item
  投保年龄：x岁
\item
  给付方式：年初给付
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
\[
\ax**{\endowxn}
\]
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{itemize}
\tightlist
\item
  保险责任：生存
\item
  保险期限：n年
\item
  投保年龄：x岁
\item
  给付方式：年初给付
\end{itemize}
\end{column}

\begin{column}{0.48\textwidth}
\[
\begin{aligned}
&\ax**{\endowxn} \\
&=\sum_{k=0}^{n-1}v^k\px[k]{x}
\end{aligned}
\]
\end{column}
\end{columns}
\end{frame}

\end{document}
